Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Use a formula to find the sum of each series. $$\sum_{i=1}^{4} 2^{i}$$

Step-by-Step Solution

Answer

The sum of the series is 30.

Step 1: Identify the Series Type

The series given is a geometric series where the common ratio is 2 and the first term is also 2, following the general form of \( a, ar, ar^2, ar^3, ... \). Here, \( a = 2 \) and the series is \( 2^1, 2^2, 2^3, 2^4 \).

Step 2: Apply the Geometric Series Formula

For a finite geometric series, the sum \( S_n \) can be found using the formula \[ S_n = a \frac{r^n - 1}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 3: Substitute the Known Values

In this case, \( a = 2 \), \( r = 2 \), and \( n = 4 \). Substitute these into the formula: \[ S_4 = 2 \frac{2^4 - 1}{2 - 1} \]

Step 4: Calculate the Expression

Calculate the expression \[ 2^4 = 16 \] Substitute this back into the formula: \[ S_4 = 2 \frac{16 - 1}{1} \] This simplifies to: \[ S_4 = 2 \times 15 = 30 \]

Step 5: Final Answer

The sum of the series \( \sum_{i=1}^{4} 2^i \) is \( 30 \).

Key Concepts

Sum of Series

Understanding the sum of a series is crucial in dealing with sequences and series in mathematics. When we talk about the 'sum of a series,' we are essentially discussing the total value when all the terms in that specific series have been added together. Geometric series, like the one in this exercise, follow a specific formula that helps simplify this summing process.

For a geometric series, the sum can be calculated using the formula:

• \( S_n = a \frac{r^n - 1}{r - 1} \)

where:

• \( S_n \) represents the sum of the series,
• \( a \) is the first term,
• \( r \) is the common ratio,
• \( n \) is the total number of terms.

This formula is powerful because it allows you to compute the sum without individually adding all terms. The specifics of this formula derive from the properties of geometric sequences. These sequences have terms that increase or decrease by a constant multiplication, making this formula possible to apply.When solving geometric series sum problems, always ensure that the series is truly geometric and not arithmetic or another type. This method of calculating the sum is only valid when the series holds the properties of a geometric progression.

Common Ratio

The 'common ratio' is a fundamental component of a geometric series. It is essentially the number by which you multiply each term to get the next term in the series. Identifying the common ratio is important for correctly using the series sum formula.

For example, in the assigned series \( 2^1, 2^2, 2^3, 2^4 \), the common ratio \( r \) is \( 2 \). This means each term is multiplied by \( 2 \) to yield the subsequent term. The identifying feature of the common ratio is its constancy throughout the sequence. The series maintains its geometric property due to this fixed multiplier.

To find the common ratio in any geometric series:

• Choose any two consecutive terms in the series.
• Divide the second term by the first term.

In a formulaic form, if \( a_2 \) is the second term and \( a_1 \) is the first term, then \( r = \frac{a_2}{a_1} \).

The role of the common ratio extends beyond mere identification; it affects the series' growth rate and helps determine the sequence's behavior over a span of terms.

Finite Series

In mathematics, a 'finite series' refers to a series that has a limited number of terms. These are particularly nice to deal with because they have a clear beginning and end, which simplifies calculations for sums using established formulas.

The exercise you encountered deals with a "finite geometric series," which means it includes a specific finite number of terms originally from a geometric progression. The term 'finite' distinguishes these from infinite series where the terms continue indefinitely.

In working with finite series, especially geometric ones, the following aspects are noteworthy:

• Finite series have a predetermined number of terms, like \( n = 4 \) in your problem.
• Calculated sums using a formula provide accurate totals without manually adding each term.
• These formulas rely on clearly defined series start and endpoint, such as the first term \( a \) and the common ratio \( r \).

Finite series are practical in educational settings because they are more manageable than infinite series. Their practicality lies in their ease of solving with direct calculation methods, making geometric progression exercises less daunting and more straightforward to tackle.